Related papers: A Mean Value Theorem for general Dirichlet Series

Several results are obtained concerning the function $\Delta_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding…

$L$ functions based on Dirichlet characters are natural generalizations of the Riemann $\zeta(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime…

We develop the theory and properties of primitive unimodular $S$-arithmetic lattices in $\mathbb{Q}_S^d$ by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas.…

The main purpose of this paper is to establish bounds on the second moment of $L\big(\tfrac{1}{2}+it,\chi\big)$, averaged over families of fixed order characters. A discrete version of the main result is also stated, from which zero-density…

Let $L(s, \chi_1), \ldots, L(s, \chi_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(\rho,\chi_1)|+\dots+a_N\log|L(\rho,\chi_N)|$…

The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$. We prove that $\sum_{n\le x} \Delta(n) \ll x(\log\log x)^{11/4}$ for all $x\ge100$. This improves on…

First we prove a modified version of the famous Lemma on the mean square estimate for exponential sums, by plugging the Cesaro weights in the right hand side of Gallagher's inequality. Then we apply it, in order to establish a mean value…

We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of…

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…

We provide upper bounds for the mean square integral $$ \int_X^{2X}(\Delta_k(x+h) - \Delta_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $\Delta_k(x)$…

For Dirichlet series roughly of the type $Z(s, w) = sum_d L(s, chi_d) d^{-w}$ the subconvexity bound $Z(s, w) \ll (sw(s+w))^{1/6+\varepsilon}$ is proved on the critical lines $\Re s = \Re w = 1/2$. The convexity bound would replace 1/6 with…

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor n\theta+\alpha \rfloor) \beta^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $\alpha$ is a real number, $\theta$ is an irrational real…

We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In…

We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll…

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is…

The research in the subfield of analytic number theory around error term of summation of sigma functions possesses a history which can be dated back to the mid-19th century when Dirichlet provided an $O(\sqrt{n})$ estimation of error term…

First we reprove, using representation theory and the relative trace formula of Jacquet, an average value result of Duke for modular L-series at the critical center. We also establish a refinement. To be precise, the L-value which appears…

In this paper we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the…

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the…